A matrix-algebraic formulation of distributed-memory maximal cardinality matching algorithms in bipartite graphs

Abstract

We describe parallel algorithms for computing maximal cardinality matching in a bipartite graph on distributedmemory systems. Unlike traditional algorithms that match one vertex at a time, our algorithms process many unmatched vertices simultaneously using a matrix-algebraic formulation of maximal matching. This generic matrix-algebraic framework is used to develop three efficient maximal matching algorithms with minimal changes. The newly developed algorithms have two benefits over existing graph-based algorithms. First, unlike existing parallel algorithms, cardinality of matching obtained by the new algorithms stays constant with increasing processor counts, which is important for predictable and reproducible performance. Second, relying on bulk-synchronous matrix operations, these algorithms expose a higher degree of parallelism on distributed-memory platforms than existing graph-based algorithms.We report high-performance implementations of three maximal matching algorithms using hybrid OpenMP-MPI and evaluate the performance of these algorithm using more than 35 real and randomly generated graphs. On real instances, our algorithms achieve up to 200× speedup on 2048 cores of a Cray XC30 supercomputer. Even higher speedups are obtained on larger synthetically generated graphs where our algorithms show good scaling on up to 16,384 cores.

Combinatorial BLAS 2.0: Scaling combinatorial algorithms on distributed-memory systems

“…Vector replications to avoid asynchronous communication in graph matching algorithms. CombBLAS powers distributed-memory maximal cardinality matching [22], maximum cardinality matching [7] and heavy-weight perfect matching [17] algorithms for bipartite graphs. All of these algorithms iteratively match vertices between two parts of a bipartite graph, where endpoints of currently matched edges are stored in two dense vectors (instances of FullyDistVec).…”